In total, we have 21 galaxies at \(z=2.86\) –\(3.77\)  and \(z=3.91\) –\(4.44\)  with \(F775W\le 27.5\)  mag as the parent sample listed in Table REF . An overview of the 21 sources is given in Figures REF  to REF  (see Figure REF  for enlarged panels for two sources as examples). We calculated the rest-frame UV magnitude (\(M_{1500}\) ) and the UV slope (\(\beta \) ) using \(z_{\rm s}\) , \(F775W\) , \(F850LP\) , and \(F105W\)  in the same manner as in [1]}. The \(M_{1500}\)  ranges from \(-19.9\)  to \(-18.1\)  with an average value of \(-18.9\) , which is fainter than \(M^*\)  at \(z\simeq 3.7\) , -20.88 mag [2]}. The average \(M_{1500}\)  corresponds to a typical dark matter halo mass of \(M_{\rm h}\simeq 1\times 10^{11}\) –\(2\times 10^{11}\ {\rm M}_\odot \) , which is estimated from a \(M_{1500}\) –\(M_{\rm h}\)  relation from the GALICS semi-analytic model in [3]}. The UV slopes range from \(-2.3\)  to \(0.2\) , with an average value of \(-1.6\) . The distribution of \(M_{1500}\)  and \(z_{\rm s}\)  of the sample is shown in Figure REF c. Out of the 21 galaxies, six galaxies, RID=7067, 7901, 9814, 9863, 10018, and 23124 (MID=7091, 180, 149, 106, 6700, and 7073) were included in the \(\mbox{Ly$\alpha $}\) -selected sample for the previous MUSE-LAH study in the HUDF, and all of them are confirmed to be LAHs [4]}. Among the 15 new sources, five galaxies, RID=5479, 7876, 22230, 23408, and 23839 (MID=7089, 103, 163, 174, and 118), are included in the MUSE-HUDF catalog in [5]} as well as in the updated version in Bacon et al. (in prep.), but they were not selected in the sample in [4]}, due to their low ZCONF values (\(\le 1\) ), low S/N values of \(\mbox{Ly$\alpha $}\)  (\(<6\) ), or close neighboring objects. In order to validate that our sample is not biased s the LAE selection, we checked LAE fractions (\(X_{\rm LAE}\) ) for our sample following [1]}. Rest-frame equivalent widths of \(\mbox{Ly$\alpha $}\)  emission (\(EW(\mbox{Ly$\alpha $})\) ) were calculated with \(\mbox{Ly$\alpha $}\)  fluxes at the galaxy's stellar-component scale (see Appendix REF  for a description of EW measurements and \(X_{\rm LAE}\)  calculations). With \(EW(\mbox{Ly$\alpha $})\ge 20\)  Å, which is a common criterion of LAEs, the \(X_{\rm LAE}\)  for our entire sample (\(-20\le M_{1500}\le -18.0\) , \(z=2.9\) –\(4.4\) ) is \(0.33^{+0.11}_{-0.09}\) . For a fair comparison, we also calculated the \(X_{\rm LAE}\)  with \(EW(\mbox{Ly$\alpha $})\ge 65\)  Å for our entire sample as \(0.14^{+0.09}_{-0.06}\) , which is similar to \(X_{\rm LAE}=0.04^{+0.02}_{-0.01}\) , \(0.07^{+0.04}_{-0.02}\) , and \(0.11^{+0.07}_{-0.04}\)  for \(EW(\mbox{Ly$\alpha $})\ge 65\)  Å and \(-21.75\le M_{1500}\le -17.75\)  at z=3.3, 4.1, and 4.7, respectively, in [1]}. It suggests that our sample is unbiased. The small difference in the \(X_{\rm LAE}\)  between the two samples could be explained by the cosmic variance due to our small survey volume in the MXDF (\(4\times 10^3\)  cMpc\(^3\) ).
